46 - GANESHPEASAD, 



is finite and continuous; f"{x) = cos when x is different from zero, 



and is non-existent at x = 0. 



d 



Now it is easily seen that, as t approaches zero, — V (0, t) behaves as 



,2 



1 r'' - — 



^ff"{x')e '*dx'. 



\7tt 0 



d 



And, therefore, it foUows, from Gase I of Art. 21, that lim V(0, t) does 



< = + o öc 



not exist. 



Therefore , excepting tc and — % , the only point where any discrepancy 

 occurs is X = 0. Here not only is t "{x) non-existent and discontinuous but 

 ö 



also F(0, t) makes an infinite number of finite oscillations as t approaches zero. 



(v) Let T{x,0) ■= f[x) = f dxf f\{x)dx, f\{x) being the f\x) in (iii) of 



'o ' 0 



Art. 24, Then f'{x) as well as f (x) is finite and continuous. At every point 

 of the everywhere dense aggregate G^, f" (x) is non-existent and has a discon- 

 tinuity of the second kind ; at all other points f^ix) = f"{x) and is finite and 

 continuous. 



Therefore at any point of (r,, (A) is meaningless. Here not only is f"{x) 



d 



non-existent and discontinuous but also V{x, f) makes an infinite number of 



dt 



finite oscillations as t approaches zero. 



(vi) Let T{x,0) = f{x) = /''cos 0<x<jt. Then f{x) is finite 



and continuous. At x — 0, f [x) is non-existent and has a discontinuity of the 

 second kind ; at all other points /' [x) as well as /' " {x) exists and is finite and 

 continuous. 



ö 



Now, it is easily seen tbat, as t approaches zero, ^(OjO behaves as 



And, therefore, it follows from Art. 17 that 1^ F(0,*)J±=- 1. 



Therefore, excepting n and — jt, the only point where any discrepancy occurs 

 is X = 0. Here not only is f"{x) non-existent but also F(0, i!)±^l. 



